Power distribution system configured as a radial network

ABSTRACT

A power distribution system configured as a radial network includes buses having respective voltages, and distribution lines having respective currents. The radial network interconnects the buses with the distribution lines in a tree-like manner. A bus has a link to at least two distribution lines. The bus voltages and distribution line currents are determined by a processing circuitry configured to receive a Branch Matrix (BM), iteratively determine currents for the distribution lines and voltages for each of the buses until a difference is below a predetermined tolerance, and output final bus voltages and final distribution line currents. The circuitry iteratively determines the currents by determining a current matrix (CM) using the BM, and by determining the currents for the plurality of distribution lines in a zig zag manner over the matrix elements in the CM. The system finds a solution using fewer iterations than the backward forward sweep method.

BACKGROUND Technical Field

The present disclosure is directed to a method and system fordetermining the load flow for distribution systems, in particular amethod of determining the load flow for extended radial distributionsystems where multi-terminal lines exist, the method uses zig zag basedload flow in a nodal current matrix.

Description of the Related Art

The “background” description provided herein is for the purpose ofgenerally presenting the context of the disclosure. Work of thepresently named inventors, to the extent it is described in thisbackground section, as well as aspects of the description which may nototherwise qualify as prior art at the time of filing, are neitherexpressly or impliedly admitted as prior art against the presentinvention.

The electric power energy chain consists of four main parts; generationplants, transmission grid, distribution grid and end-user.

Power plant consists of power generation units of different types undertwo main categories; 1) conventional (e.g. fossil fuel-based units) andrenewables (e.g. solar and wind fuel-based units). Power generationplants could be centralized or distributed and the overall system set upcaters for operation, maintenance and emergency scenarios. FIG. 1 is adiagram of an exemplary power grid. The electric power is usuallygenerated at alternating current (AC) using 3-phase configuration. Thepower leaves the generator 101 and enters a transmission substation 103.

Transmission grid is for long distance transmission and typicallytransmits at high voltages (above 69 kV). The transmission substation103 uses transformers to convert the generator's 101 voltage to highvoltage levels for long-distance transmission on a transmission grid105.

The power that comes off the transmission grid 105 is stepped-down to adistribution grid 120. The distribution grid distributes power to homesand businesses at lower voltage level (69 kV and below) for shortdistances. The distribution grid 120 may step down voltages in severalphases in power substations. A main, or bulk, power substation 111 maydistribute power to other distribution substations 115. A distributionsubstation 115 may have transformers that step-down voltages for furtherpower distribution at different voltages. One group of distributionlines 121 may include a transformer 119 to step the power down to astandard line voltage for the set of lines. Another group ofdistribution lines 121 may leave the distribution substation 115 at ahigher voltage. Distribution lines 121 that carry high voltages will bestepped down further before entering residential buildings andbusinesses. This further stepping-down occurs in other substations or inother transformers 123. Distribution lines 121 running near a house willbe reduced from 13800 volts down to 240 volts that makes up a normalhousehold electric service.

In addition to consuming power from the electric power grid, consumersmay also generate their own power while being linked to the power grid.Provided a special electrical meter and current inversion equipment, aconsumer can obtain power from renewable energy sources such as solarpanels and wind turbines. As a consumer generates more power that theyneed, that excess power may be a power generation source for the powergrid. This concept falls under distributed generation.

The electric power distribution system 120 is the final stage in thedelivery of electric power. It carries electricity from a transmissionsystem 105 to individual consumers 125. Distribution lines 121 may carrya medium voltage power ranging between 1 kV and 35 kV to distributiontransformers 123 which are located near the consumer's premises.Distribution transformers 123 further lower the voltage to the voltageof household appliances and may feed several consumers through secondarydistribution lines at this voltage.

Power distribution systems in sparsely populated areas may be configuredwith a multi-terminal line. As an example, in some sparsely populatedareas tapped lines may exist within power distribution networks to savethe cost of building a substation at connection points in rural areas.Customers may also have local generation which may contribute to thenetwork. A multi-terminal line is defined as a power line with three ormore terminals. FIG. 2 illustrates a three terminal transmission line. Amulti-terminal line may have a single circuit line 201 supplying aseries of customers, typically in remote communities. The line 201 mayhave either a load 231, 233 or a power generation 221, 223, 225 or bothat any terminal. Each terminal may be connected to an associatedtransformer 241, 243, 245 and bus 211, 213, 215.

Distribution networks for power distribution systems are of two majortypes depending on the arrangement of their distribution lines: radialor network. Some distribution network arrangements may be less than afull network, referred to as a weakly meshed structure. A weakly mesheddistribution network contains some loops whereas a radial network iswithout loops. A radial system is arranged like a tree where eachcustomer has one source of supply. A network system has multiple sourcesof supply operating in parallel. A majority of power distributionsystems for sparsely populated areas and remotely scattered hydrocarbonfacilities have a radial structure configuration.

As stated above, a distribution system originates at a substation wherethe electric power is converted from the high voltage transmissionsystem to a lower voltage for delivery to the consumers. A radialdistribution system is one in which separate distribution lines, alsoknown as feeders, radiate from a single substation and feed thedistributors at only one end. For simplicity, a radial distributionsystem may be represented as a line diagram. FIG. 3 is an exemplary linediagram for a radial distribution network. The line diagram for theradial distribution network has n nodes (representing buses), b branches(representing transmission lies) and a single voltage source(representing the substation connected to the transmission grid) at theroot node. A node (bus, or bus bar) is for local current powerdistribution and distributes power to distribution lines, which fan outto customers. A branch is a distribution circuit line section between apair of nodes. In this tree structure, a branch numbering scheme may beused in which the node of a branch L closest to the root node is L1 andthe farther node is L2. In other words, a branch starts from a sendingnode (bus at the root side) and ends at an ending node (bus). A radialdistribution network leaves a single station and passes through thenetwork area with no normal connection to any other supply. In thisradial delivery network each node is connected to the substation via atleast one path. Power flow in the radial distribution network is suchthat the power is delivered from the main branch to the sub branchesthen is split out from the sub-branches, where the power is transferredfrom root node and is split. Power flow in the radial distributionnetwork has no loops and each bus is connected to the source via exactlyone path.

Power grids are subjected to great demands due to expansions of thepower networks. Rapid development of communities and businesses leads toincreased power transmission requirements. The expansion can beaccomplished by adding new distribution lines and by upgrading existinglines by adding new devices. The expansion must be made whilemaintaining a stable power transmission network. Voltage instability inthe network may lead to system collapse, when bus voltage drops to sucha level from which it cannot recover. In such a situation, completesystem blackouts may take place. Voltage stability analysis is performedto ensure expansion of a network is successful and prevents system loss.Load Flow or Power Flow Study and Analysis is performed as part ofvoltage stability studies and contingency analysis.

To begin a voltage stability analysis of a power system, complexvoltages at all busses are determined. After this, power flows from, abus and the power flowing in all the transmission lines are determined.Such power flow is performed using load flow analysis. Load flowanalysis determines steady state voltage magnitudes at all buses, for aparticular load condition. Load flow analysis may be used in planningstudies, for designing a new network or expansion of an existingnetwork. Calculated values of load flow and voltage may be compared withthe steady state device limits in order to estimate the health of thenetwork.

Load flow analysis represents the first building block toward any powersystem related study. This includes energy forecast, voltage drop andstability analysis in its three parts; steady state, dynamic andtransient. The latter requires load flow to calculate the initialconditions.

Load flow or power flow analysis is performed on power distributionsystems to understand the nature of the installed network. Load flowanalysis is performed as a computational procedure to determine thesteady state operating characteristics of a power system network fromgiven line data and bus data, i.e., a given loading. The output of aload flow analysis is nodal voltages and the phase angles, real andreactive power (both sides of each line), power flows and the linelosses in a network. Load flow analysis involves finding all possiblenode voltages. A load flow study may be used to estimate a best locationfor a new generation station, substation and new distribution lines.

Power and voltage constraints when conducting a load flow study imposenon-linearity that require iterative techniques to solve. The problem isnon-linear because the power flow into load impedances is a function ofthe square of the applied voltages. Due to the non-linear nature of thisproblem, numerical methods are employed to obtain a solution that iswithin an acceptable tolerance.

There are several different methods of solving nonlinear system ofequations. Conventional methods include Newton-Raphson method,Gauss-Seidel method, and Fast Decoupling load flow method. TheNewton-Raphson method begins with initial guesses of all unknownvariables (voltage magnitude and angles at Load Buses and voltage anglesat Generator Buses). A Taylor Series is produced for each of the powerbalance equations included in the system of equations. The system ofequations is solved to determine a next guess of voltage magnitude andangles. The process continues until a stopping condition is met.Typically, the stopping condition is to terminate when the mismatchequations are below a specified tolerance. The Gauss-Seidel method isanother iterative method that uses little memory and does not solve amatrix. However, the Gauss-Seidel method is slower to converge thanother iterative methods. The Fast Decoupling method uses an approximatedecoupling of active and reactive flows in well-behaved power networks.The Fast Decoupling method converges faster than the Newton-Raphsonmethod and has been used for real-time management of power grids.

Distribution networks fall into the category of ill-conditioned.Features that relate to an ill-conditioned network include that thenetwork is radial or weakly meshed, a network that has a high R/X ratio,the network performs multi-phase, unbalanced operation, the load for thenetwork is unbalanced and distributed, or generation is distributed. TheX/R ratio is the ratio of the system reactance to the system resistance,looking back to the power source from a point in a power circuit. Thus,the X/R ratio is dependent on resistance of a power line.

The majority of power flow algorithms used in industry are based on theNewton-Raphson method and its variants. Conventional methods includingthe Newton Raphson, Fast Decoupling and Gauss Seidel methods are bestsuited for well-behaved power distribution systems. The Newton-Raphsonmethod and its variants have been developed specifically fortransmission systems which have a meshed structure, with parallel linesand many redundant paths from the generation points to the load points.

However, the Newton-Raphson method and related transmission algorithmshave failed with ill-conditioned distribution networks. The FastDecoupling Load Flow method and the Gauss Seidel power flow techniquehave been found to be inefficient in solving ill-conditioned powersystems. Their computational time is long. See J. A. M. Rupa and S.Ganesh, “Power Flow Analysis for Radial Distribution System UsingBackward/Forward Sweep Method,” vol. 8, no. 10, pp. 1537-1541, 2014,incorporated herein by reference in its entirety.

A backward/forward sweep (BFS) method has been proposed forill-conditioned distribution networks, particularly those arranged as aradial distribution network. Unlike Newton-Raphson methods, thebackward/forward sweeping method does not need a Jacobian matrix. In thebackward/forward sweep method, the power of each branch is calculatedusing backward propagation and the voltage magnitudes at each node arecalculated in forward propagation. The backward/forward sweep method hasbeen found to have fast convergence and is suitable for radialstructure.

The Backward/Forward Sweep (BFS) power flow algorithm was developed as atechnique capable of solving radial and weakly meshed distributionnetworks with up to several thousand line sections (branches) and nodes(buses), and that is robust and efficient. The BFS technique isbranch-oriented instead of using nodal solution methods such asNewton-Rapson and related techniques. Also, in order to use the BFStechnique, a weakly meshed distribution network may be converted to aradial network with no loops. Maximum real and reactive power mismatchesat the network nodes may be used as a convergence criteria. See D.Shirmohammadi, H. W. Hong, A. Semlyen, and G. X. Luo, “Acompensation-based power flow method for weakly meshed distribution andtransmission networks,” IEEE Trans. Power Syst., vol. 3, no. 2, pp.753-762, May 1988 incorporated herein by reference in its entirety. Seealso M. E. Baran and F. F. Wu, “Network reconfiguration in distributionsystems for loss reduction and load balancing,” IEEE Trans. PowerDeliv., vol. 4, no. 2, pp. 1401-1407, April 1989; Xiaofeng Zhang, F.Soudi, D. Shirmohammadi, and C. S. Cheng, “A distribution short circuitanalysis approach using hybrid compensation method,” IEEE Trans. PowerSyst., vol. 10, no. 4, pp. 2053-2059, November 1995, which areincorporated herein by reference in their entirety.

The backward sweep is a calculation of the power flow through thebranches starting from the last branch and proceeding in the backwarddirection towards the root node. The forward sweep is a calculation ofthe voltage magnitude and angle of each node starting from the root nodeand proceeding in the forward direction towards the last node.

In particular, the backward sweep starts from the branches in the lastlayer and updates effective power flow in each branch by considering thenode voltages of a previous iteration. The forward sweep is a voltagedrop calculation with possible current or power flow updates. Nodalvoltages are updated starting from branches in the first layer towardthose in the last layer. A purpose of the forward propagation is tocalculate the voltages at each node starting from the feeder sourcenode. During the forward propagation the effective power in each branchis held constant to the value obtained in backward sweep. Also, voltagevalues obtained in the forward path are held constant during thebackward propagation and updated power flows in each branch aretransmitted backward along the feeder using the backward path.

Rupa has developed one approach for performing the backward/forwardsweep method for power flow analysis in radial distribution systems. Theapproach in Rupa addresses a problem that the conventional backwardforward sweep method is not useful for modern active distributionnetworks. That method solves a recursive relation of voltage magnitudes.In particular, the method solves recursive relations that use realpower, reactive power, and voltage magnitude at each branch. The methodcan be used to obtain the power losses.

Several references discuss BFS and applied modifications for differentpurposes. Ju et al. describes a proposed solution that relates todistributed energy resources, in particular distributed generators. Theproposed solution is an extension of BFS that solves a problem thatconvergence deteriorates as the number of PV nodes increases, and mayeven diverge for large-branch R/X ratios. The proposed solution includesa method for handling PV nodes based on loop analysis incorporated intothe BFS technique, where PV nodes refer to nodes connected bydistributed generators with constant voltage control. The method of thepresent disclosure incorporates both Newton's method and the BFS method.In particular, a PV type node (as opposed to a PQ type node) isformulated as a Vθ bus for the DFS procedure, then a correction ofvoltage angle deviation is obtained with accurate correction equationsby the Newton method. See Y. Ju, W. Wu, B. Zhang, and H. Sun, “AnExtension of FBS Three-Phase Power Flow for Handling PV Nodes in ActiveDistribution Networks,” IEEE Trans. Smart Grid, vol. 5, no. 4, pp.1547-1555, Jul. 2014—incorporated herein by reference in its entirety.

G. Chang et al. describes a simplified BFS approach that is faster thanconventional BFS. Chang et al. describes an approach in which, unlikethe commonly used iterative load flow method, uses the real andimaginary decomposition of bus voltages, branch currents and systemimpedances in terms of real-number quantities. See G. Chang et al.adopted the linear proportion concept to calculate the bus voltages byidentifying the real to imaginary ratio of the voltage. See G. Chang, S.Chu, and H. Wang, “A Simplified Forward and Backward Sweep Approach forDistribution System Load Flow Analysis,” in 2006 InternationalConference on Power System Technology, 2006, no. 4, pp. 1-5,incorporated herein by reference in its entirety.

J. Wang et al. describe applying a BFS method and a hybrid particleSwarm method to reconfigure distribution networks with distributed powergenerations. Wang et al. describe using a node-layer incident matrix tofind the load flow. See J. Wang, L. Lu, J.-Y. Liu, and S. Zhong,“Reconfiguration of Distribution Network with Dispersed Generators Basedon Improved Forward-Backward Sweep Method,” in 2010 Asia-Pacific Powerand Energy Engineering Conference, 2010, vol. 2015—Septe, no. 1, pp.1-5, incorporated herein by reference in its entirety.

Another distributed generator related study is described in K. Kaur etal. BFS was used to carry out base power flow analysis. See K. Kaur andS. Singh, “Optimization and comparison of distributed generator indistribution system using backward and forward sweep method,” in 20167th India International Conference on Power Electronics (IICPE), 2016,pp. 1-5, incorporated herein by reference in its entirety.

A modified BFS technique for unbalanced radial distribution system isintroduced in Samal et al. See P. Samal and S. Ganguly, “A modifiedforward backward sweep load flow algorithm for unbalanced radialdistribution systems,” in 2015 IEEE Power & Energy Society GeneralMeeting, 2015, vol. 2015—Septe, pp. 1-5, incorporated herein byreference in its entirety. The method considers the mutual couplingbetween the phases to obtain higher accuracy. The method utilizes threematrices A, B and C to obtain load flow solutions while identifyingbuses. Matrix A is formed to know the downstream buses connected to aparticular bus. Matrix B is formed to identify the end buses. Matrix Cis formed to obtain actual branch currents. The three matrices are usedin identifying downstream buses and in calculating branch current orpower flow in an easy manner.

G. Setia et al. describe an improved fast decoupled method and animproved BFS method for solving power flow for distribution network. Theimproved fast decoupled method (Axis Rotation Fast Decoupled Load FlowARFDLF) uses axis rotation to converge equally good for high R/Xnetworks as low ones and to solve load flow in distribution systems. Theimproved BFS uses a new data structure. See G. A. Setia, G. H. M.Sianipar, and R. T. Paribo, “The performance comparison between fastdecoupled and backward-forward sweep in solving distribution systems,”in 2016 3rd Conference on Power Engineering and Renewable Energy(ICPERE), 2016, pp. 247-251, incorporated herein by reference in itsentirety.

Several implementations that solve load flow in distributed systems havebeen described that use algebraic matrix techniques. For example, USPatent Application Publication 2013/0289905 to Ou describes adistribution power flow analysis method. Ou explains that conventionaldistribution power flow analysis methods require substantial changes inan impedance matrix when a new node or impedance is added. As asolution, Ou describes a distribution power flow analysis system thatincludes a first relationship matrix and a second relationship matrix.The first relationship matrix indicates a relationship between a nodeinjection current matrix I and a branch current matrix B. The secondrelationship matrix indicates a relationship between a node mismatchmatrix ΔV (mismatch between a reference voltage and voltages of othernodes) and the branch current matrix B. The branch current matrix B isformed by a plurality of branch currents among nodes. The firstrelationship matrix is an upper triangular matrix only containing 0and 1. The second relationship matrix is the impedance among nodes, andis a lower triangular matrix.

The method in Ou is applicable to cases of adding a new node, impedanceor parallel loop. In particular, Ou explains that when a new node orimpedance is added, the first relationship matrix and the secondrelationship matrix are updated. The first relationship matrix isupdated by adding a new column and a new row to the first relationshipmatrix, which may be accomplished by adding a 1 in the newly addeddiagonal position and a 0 in the other new positions. The secondrelationship matrix is updated by adding a new column and a new row,which may be accomplished by adding a new impedance in the newly addeddiagonal position and a 0 in the other new positions. The methoddescribed by Ou is compared to a conventional Gauss implicit Z-matrixmethod and a Newton-Raphson method.

R. Prenc, et al. is concerned with load flow calculation for radialdistribution networks with a large number of nodes. Prenc et al.describes a modification of the backward/forward sweep method thatrepresents distribution lines by a detailed 7C equivalent model, whichdoes not neglect parallel capacitance. The proposed algorithm assumes abalanced three phase power distribution network. The proposed algorithmcovers the influence of dispersed generation which can be placed at anynode on a primary substation feeder.

An implementation of the algorithm in Prenc et al. includes forming anincidence matrix IM, where rows represent branches (lines) of thenetwork, and columns represent nodes. Every matrix element in the IM hasa value of 1, when j is the receiving node of branch i, a value of −1,when j is the sending node of branch i, and a value of 0 otherwise. Thealgorithm transforms the IM to an inverse matrix INV. The algorithmforms a new matrix BR, where rows represent branches of the network, andmatrix elements represent the ordinal number of nodes that are fed byeach branch. The matrix BR is used in a branch active/reactive powerflow, which is the backward sweep of the modified algorithms.

Together with a forward sweep to calculate node voltages, the algorithmcan be applied on any radial distribution network configuration withdispersed generation units connected simultaneously in any node of thenetwork. See R. Prenc, D. Skrlec, and V. Koman, “A novel load flowalgorithm for radial distribution networks with dispersed generation,”December 2013.

G. Meerimatha, et al. describes a method of analyzing the load flow inbalanced radial distribution systems, and in particular a techniquewhich takes the topological characteristics of radial distributionsystems and solves the system load directly by using a single businjection branch injection branch current (BIBC) matrix in forwardbackward sweep method. The BIBC matrix represents the relationshipbetween bus current injections and branch currents. The method describedin Meerimatha converges faster than an alternative method that uses twomatrices, i.e., BIBC and branch-current to bus voltage (BCBV) matrices,for solution of load flow. See G. Meerimatha, G. Kesavarao, N.Sreenivasulu, “A novel distribution system power flow algorithm usingforward backward matrix method,” in Journal of Electrical andElectronics Engineering, Vol. 10, Issue 6 Ver. II, November-December2015, pp. 46-51.

In Meerimatha, backward propagation includes forming the bus injectionto branch current matrix and finding injected load currents. Forsolution of load flow, to determine line flows of a network the branchcurrent matrix is calculated based on BIBC and injected load currents.To determine bus voltages, the forward propagation step determines thebus voltage vector. The real and reactive power loss can be calculatedin each branch using the load flow solution.

D. Ravi, et al. describes a method for handling load flow in a radialdistribution system. Ravi proposed a simplified approach to an algorithmthat requires formation of bus-injection to branch current (BIBC) matrixand branch-current to bus voltage (BCBV) matrix with primitiveimpedances as elements and distribution load flow (DLF) matrix. The DLFmatrix in the algorithm had been obtained as a product of BIBC and BCBVmatrices. Ravi proposes an improved technique that uses the distributionload flow (DLF) matrix without requiring the BIBC and BCBV matrices,which take up a large amount of memory space. Ravi noticed features ofthe DLF matrix and used those features in developing an algorithm thatuses a branch path (K) matrix. In particular, the algorithm proposed byRavi determines the elements of the DLF matrix by comparing rows andcolumns of a branch path (K) matrix. The K matrix is a combination of0's and 1's, where the 1's represent information about connecting pathbetween node-1 and any selected node. The diagonal elements of DLFmatrix are found based on the K matrix. Off diagonal elements of the DLFmatrix are determined based on a comparison of rows of the K matrix. Theproposed method in Ravi that does not involve a BIBC matrix and a BCBVmatrix has a greatly reduced memory requirement, especially for largedistribution systems. See D. Ravi, Dr. V. U. Reddy, D. P. Reddy, “Loadflow analysis for unbalanced radial distribution systems,” inInternational Journal of Electrical Electronics & Computer ScienceEngineering, Vol. 5, Issue I, February 2018, pp. 18-23.

It is one object of the present disclosure to describe a method fordetermining the load flow for extended radial distribution systems wheremulti-terminal lines exist. Other objectives include a method for loadflow analysis that reduces the number of iterations to converge andoffers superior accuracy over a range of bus distribution systems,particularly as compared to the BFS approach. In some aspects, themethod involves a Zig Zag-based load flow analysis that calculatesinitial conditions for an energy forecast, voltage drop or stabilityanalysis.

SUMMARY

An exemplary embodiment relates to a system and method for load flowanalysis of a power distribution system configured as a radial networkhaving a plurality of buses, a plurality of power distribution lines,and at least one of the buses has a link to at least two of the powerdistribution lines. The method performed by processing circuitryconfigured to receive a Branch Matrix (BM) having a row for each powerdistribution line and a column for each bus, wherein the BM containsvalues indicating buses and associated power distribution lines;iteratively determine currents for the plurality of power distributionlines and voltages across each of the buses until a difference between abus voltage and a previous iterated bus voltage at each bus is below apredetermined tolerance; and display a graph of bus voltages for eachbus. The circuitry iteratively determines the currents by determining aninitial current matrix (CM′) using the BM by multiplying a node currentwith corresponding matrix elements in the BM, wherein the node currentis a ratio of a complex power at a bus and a voltage of the bus; anddetermining the currents for the plurality of power distribution linesin a zig zag manner over the matrix elements in the CM′ and records thedetermined currents in a final current matrix (CM). The circuitryiteratively determines the voltages by determining the voltage acrosseach bus based on a voltage drop between buses, starting from busesupstream and moving to buses downstream in the direction of voltageflow.

The foregoing general description of the illustrative embodiments andthe following detailed description thereof are merely exemplary aspectsof the teachings of this disclosure, and are not restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of this disclosure and many of theattendant advantages thereof will be readily obtained as the samebecomes better understood by reference to the following detaileddescription when considered in connection with the accompanyingdrawings, wherein:

FIG. 1 illustrates an exemplary power grid;

FIG. 2 illustrates a multi-terminal power line;

FIG. 3 is a diagram of an exemplary radial distribution system;

FIG. 4 is a block diagram of the computer system, in accordance withexemplary aspects of the disclosure;

FIG. 5 is a flowchart of a backward/forward sweep (BFS) load flowmethod;

FIG. 6 is an exemplary 5-bus circuit;

FIG. 7 is a flowchart of a Zig-Zag load flow method, in accordance withexemplary aspects of the disclosure;

FIG. 8 is a flowchart of a method for finding the currents for thebranches in a Zig Zag manner, in accordance with exemplary aspects ofthe disclosure;

FIG. 9 is a graph of a voltage profile of a 5 node radial distributionnetwork as a comparison of BFS and Zig Zag techniques;

FIG. 10 is a graph of a voltage profile of a 7 node radial distributionnetwork as a comparison of BFS and Zig Zag techniques;

FIG. 11 is a graph of a voltage profile of a 11 node radial distributionnetwork as a comparison of BFS and Zig Zag techniques;

FIG. 12 is a graph of a voltage profile of a 25 node radial distributionnetwork as a comparison of BFS and Zig Zag techniques;

FIG. 13 is a graph of a voltage profile of a 28 node radial distributionnetwork as a comparison of BFS and Zig Zag techniques;

FIG. 14 is a graph of a voltage profile of a 30 node radial distributionnetwork as a comparison of BFS and Zig Zag techniques;

FIG. 15 is a graph of a voltage profile of a 33 node radial distributionnetwork as a comparison of BFS and Zig Zag techniques;

FIG. 16 is a graph of a voltage profile of a 34 node radial distributionnetwork as a comparison of BFS and Zig Zag techniques;

FIG. 17 is a graph of a voltage profile of a 69 node radial distributionnetwork as a comparison of BFS and Zig Zag techniques; and

FIG. 18 is a flowchart in accordance with exemplary aspects of thedisclosure.

DETAILED DESCRIPTION

In the drawings, like reference numerals designate identical orcorresponding parts throughout the several views. Further, as usedherein, the words “a,” “an” and the like generally carry a meaning of“one or more,” unless stated otherwise. The drawings are generally drawnto scale unless specified otherwise or illustrating schematic structuresor flowcharts.

Furthermore, the terms “approximately,” “approximate,” “about,” andsimilar terms generally refer to ranges that include the identifiedvalue within a margin of 20%, 10%, or preferably 5%, and any valuestherebetween.

Backward/Forward Sweep (BFS) and Zig Zag power flow techniques in thisdisclosure both achieve accurate distribution load flow results inradial distribution systems. Aspects of this disclosure are directed toa Zig Zag-based load flow approach that achieves the same or betterdistribution load flow accuracy using fewer iterations to converge thanthe BFS power flow analysis approach. The Zig Zag-based load flow isapplicable to extended radial distribution systems having multi-terminallines.

The backward/forward sweep method for the load flow computation is aniterative method in which, at each iteration, two computational stagesare performed. The first stage is for calculation of the power flowthrough the branches starting from the last branch and proceeding in thebackward direction towards the root node. The other stage is forcalculating the voltage magnitude and angle of each node starting fromthe root node and proceeding in the forward direction towards the lastnode.

A radial distribution network interconnects the buses with thedistribution lines beginning with the root bus in a tree-like manner.The number of buses between a bus and the root bus defines a layer. Forexample, a bus that is the third bus in the load direction from the rootis in a second layer of the network. The distribution lines may be ofdifferent lengths, and thus will have corresponding impedance andvoltage drop. Also, the voltage of a bus at one of the layers may bedifferent from a voltage of a bus at a different one of the layers. Anybus may have a link to at least two of the distribution lines. Theradial configuration entails that there are no connected loops in thesystem and each node is linked to the source via one way. Loops mightexist but separated via a normally open breaker. A distribution systemcombines single-, two-, and three-phase circuits, delta-connected andunbalanced loads that are considered challenging. It is the mosteconomical power supply and has an acceptable reliability index tocertain types of customers. The radial distribution configuration istypically used in sparsely populated areas and in the remote scatteredhydrocarbon facilities.

Embodiments of the present invention create a nodal current matrix and,using the nodal current matrix, find the currents for the systembranches in a Zig Zag manner. The zig zag manner of finding currents forsystem branches replaces the backward sweep stage of the BFS method andgreatly reduces the number of iterations needed for convergence.

FIG. 18 illustrates an embodiment of the present disclosure in flowchartform. FIG. 18 shows steps of a method for load flow analysis of a powerdistribution system. The method begins at S1801 that shows receipt of aBranch Matrix (BM) having a row for each power distribution line and acolumn for each bus in which the BM contains values indicating buses andassociated power distribution lines. This is followed by S1802 includingdetermining currents for the plurality of power distribution lines andvoltages across each of the buses until a difference between a busvoltage and a previous iterated bus voltage at each bus is below apredetermined tolerance. Determining the current (S1802) includes(S1803) determining an initial current matrix (CM′) using the BM bymultiplying a node current with corresponding matrix elements in the BMin which the node current is a ratio of a complex power at a bus and avoltage of the bus and (S1804) determining the currents for theplurality of power distribution lines in a zig zag manner over thematrix elements in the CM′ and records the determined currents in afinal current matrix (CM). S1805 shows determining the voltage acrosseach bus based on a voltage drop between buses, starting from busesupstream and moving to buses downstream in the direction of voltageflow. The bus voltages are displayed on a graph for each bus (S1806).

Computer System

The disclosed methods may be performed in a computer system, such as alaptop computer, desktop computer, or a computer workstation. Thedisclosed method may also be performed in a virtual computer environmentsuch as a cloud service, data center or virtualized server network.

FIG. 4 is a block diagram of a basic computer system. In oneimplementation, the functions and processes of the computer system maybe implemented by a computer 426. Next, a hardware description of thecomputer 426 according to exemplary embodiments is described withreference to FIG. 4. In FIG. 4, the computer 426 includes a CPU 400which performs the processes described herein. The process data andinstructions may be stored in memory 402. These processes andinstructions may also be stored on a storage medium disk 404 such as ahard drive (HDD) or portable storage medium or may be stored remotely.Further, the claimed advancements are not limited by the form of thecomputer-readable media on which the instructions of the inventiveprocess are stored. For example, the instructions may be stored on CDs,DVDs, in FLASH memory, RAM, ROM, PROM, EPROM, EEPROM, hard disk or anyother information processing device with which the computer 426communicates, such as a server or computer.

Further, the claimed advancements may be provided as a utilityapplication, background daemon, or component of an operating system, orcombination thereof, executing in conjunction with CPU 400 and anoperating system such as Microsoft® Windows®, UNIX®, Oracle® Solaris,LINUX®, Apple macOS® and other systems known to those skilled in theart.

In order to achieve the computer 426, the hardware elements may berealized by various circuitry elements, known to those skilled in theart. For example, CPU 400 may be a Xenon® or Core® processor from IntelCorporation of America or an Opteron® processor from AMD of America, ormay be other processor types that would be recognized by one of ordinaryskill in the art. Alternatively, the CPU 400 may be implemented on anFPGA, ASIC, PLD or using discrete logic circuits, as one of ordinaryskill in the art would recognize. Further, CPU 400 may be implemented asmultiple processors cooperatively working in parallel to perform theinstructions of the inventive processes described above.

The computer 426 in FIG. 4 also includes a network controller 406, suchas an Intel Ethernet PRO network interface card from Intel Corporationof America, for interfacing with network 424. As can be appreciated, thenetwork 424 can be a public network, such as the Internet, or a privatenetwork such as LAN or WAN network, or any combination thereof and canalso include PSTN or ISDN sub-networks. The network 424 can also bewired, such as an Ethernet network, or can be wireless such as acellular network including EDGE, 3G and 4G wireless cellular systems.The wireless network can also be WiFi®, Bluetooth®, or any otherwireless form of communication that is known.

The computer 426 further includes a display controller 408, such as aNVIDIA® GeForce® GTX or Quadro® graphics adaptor from NVIDIA Corporationof America for interfacing with display 410, such as a Hewlett Packard®HPL2445w LCD monitor. A general purpose I/O interface 412 interfaceswith a keyboard and/or mouse 414 as well as an optional touch screenpanel 416 on or separate from display 410. General purpose I/O interfacealso connects to a variety of peripherals 418 including printers andscanners, such as an OfficeJet® or DeskJet® from Hewlett Packard®.

The general purpose storage controller 420 connects the storage mediumdisk 404 with communication bus 422, which may be an ISA, EISA, VESA,PCI, or similar, for interconnecting all of the components of thecomputer 426. A description of the general features and functionality ofthe display 410, keyboard and/or mouse 434, as well as the displaycontroller 408, storage controller 420, network controller 406, andgeneral purpose I/O interface 412 is omitted herein for brevity as thesefeatures are known.

Backward/Forward Sweep

Rupa describes a backward/forward sweep method that uses an objectivefunction in terms of real and reactive power to find power flow as:

P_(k + 1) = P_(k) − P_(loss, k) − P_(Lk + 1)Q_(k + 1) = Q_(k) − Q_(loss, k) − Q_(Lk + 1)

where P_(k) is the real power flowing out of a bus; Q_(k) is thereactive power flowing out of the bus; P_(Lk+1) is the real power at busk+1; Q_(Lk+1) is the reactive power at bus k+1, where k is an identifierassigned to a bus and k+1 refers to a bus at the other end of a line inthe load direction. During backward propagation, effective real andreactive power flows of all branches are computed. During forwardpropagation, node (bus) voltages and phase angles are updated. The powerloss P_(loss,k) or Q_(loss,k) may be computed in the line sectionconnecting busses k and k+1. The total power loss of the feeder may bedetermined by summing up the losses of all line sections of the feeder.

FIG. 5 is a flowchart of a Backward/Forward Sweep method. The method inFIG. 5 has been implemented for purposes of comparison with the Zig Zagmethod of the present invention. As mentioned above, BFS is one of therobust techniques that is widely used in a distribution network analysisfor load flow in radial distribution systems. This is because of itseffectiveness for radial and weak meshed systems.

In S501, the BFS method reads network configuration data. In S503, theBFS method initializes voltage V and current I. The BFS method thenperforms the backward sweep S505 to calculate currents. The currents arecalculated across each branch, starting with the last layer in thesystem. In the backward sweep, the BFS method moves backward to thebranches connected to the Swing bus, which is usually bus no. 1. Thecurrents are calculated using the following formula:

$\begin{matrix}{I_{i} = {( \frac{S_{i}}{V_{i}} )^{*} + {\sum\limits_{{k = a},b,c,{\ldots\;{etc}}}I_{k}}}} & (1)\end{matrix}$

Where i is a specific branch in which the current is calculated. Thesymbols a, b and c, etc. represent the branches connected to Branch ifrom the load side. For purposes of this disclosure, real and reactivepower are represented as the symbol S for complex power in a per unitquantity.

Next, the BFS method performs a forward sweep S507 to calculate thevoltages. The voltage across each bus is calculated in a forward manner.In the forward sweep, the BFS method starts with the Swing bus andcovers all the other buses in the network. The calculation is performedusing the following voltage drop formula.

$\begin{matrix}{V_{i} = {V_{j} - {I_{ij}Z_{ij}}}} & (2)\end{matrix}$

Where i is a specific bus that is under calculation. The symbol j is abus with a former index (j<i) that is directly connected to i (loadside). I_(j) and Z_(ij) are the current and impedance between buses iand j.

The BFS method is performed iteratively until a convergence criteria,S509, is met. In some BFS approaches, convergence may be achieved when avoltage mismatch (difference between voltage V and a previous voltageVat bus i) is less than a tolerance of about 0.0001.

When the BFS method meets the convergence criteria (YES in S509), inS511, the computer system outputs final branch currents and final busvoltages V,. The final bus voltages may be displayed as a graph for eachbus, or a selected range of busses.

Zig Zag-Based Load Flow

Disclosed embodiments relate to an improvement over the BFS method. Ithas been determined that a specific order of processing over a matrixmay lead to fewer number of iterations to converge. In particular, thedisclosed embodiments involve processing over a matrix in a Zig Zagorder. The matrix that is processed in this manner may have severalhundred to several thousand rows and columns. The Zig Zag-based loadflow method may be implemented as a computer program in a programminglanguage such as Fortran or C, or in a numerical computing environmentsuch as MatLab, Mathematica, or scripting language such as Python, toname a few. The computer program may be executed in a computer systemsuch as that in FIG. 4.

FIG. 7 is a flowchart of a Zig Zag-based load flow method, in accordancewith exemplary aspects of the disclosure. The Zig Zag method calculatescurrents using a zig zag pattern over a nodal current matrix instead ofusing backward sweep as in the BFS method. The Zig Zag-based method isapplicable to extended radial distribution systems where multi-terminallines exist. The Zig Zag-based load flow method begins with, S701,reading in network configuration data and, in S703, initializing voltageV and current I. Configuration data involves labeling of branches andnodes in the radial distribution network. As noted above, a branch is adistribution line between two nodes. A branch will have a current flowand a voltage drop due in part to the length of a branch. A node (bus),also referred to as a bus bar, distributes power to distribution lines,which fan out to other buses or customers. The bus will have a voltage.Also, a load flow analysis may be used for designing a new distributionnetwork or expansion of an existing distribution network. The branchesand nodes represent distribution lines and buses for a new distributionnetwork or expansion in an existing network. For purposes of simplifyinga representation of a network, branches may be represented as singlelines and busses may be represented as a connection point betweenbranches.

After initialization, the Zig Zag-based method includes the followingsteps:

-   1) S705, creating a branch matrix (BM) as shown below where the rows    reflect the circuit branches and the columns represent circuit    buses.

$\begin{matrix}{{{BM} = \begin{bmatrix}B_{11} & \ldots & B_{1N} \\\vdots & \ddots & \vdots \\B_{M1} & \ldots & B_{MN}\end{bmatrix}}{B_{ij} = \{ \begin{matrix}1 & {{if}\mspace{14mu}{bus}\mspace{14mu} j\mspace{14mu}{exists}\mspace{14mu}{in}\mspace{14mu}{Branch}\mspace{14mu} i} \\0 & {{if}\mspace{14mu}{bus}\mspace{14mu} j\mspace{14mu}{does}\mspace{14mu}{not}\mspace{14mu}{exist}\mspace{14mu}{in}\mspace{14mu}{Branch}\mspace{14mu} i}\end{matrix} }} & (3)\end{matrix}$

-   -   B_(ij) represents the existence of a bus in a certain branch. It        takes either 1 (bus exists) or 0 (bus does not exist). M and N        reflects the number of branches and buses in a circuit        respectively.    -   FIG. 6 illustrates an example 5-bus circuit with five buses and        two branches. A BM matrix for the example 5-bus circuit shown in        FIG. 6, is as follows:

$\begin{matrix}{{BM} = \begin{bmatrix}1 & 1 & 1 & 1 & 0 \\0 & 0 & 1 & 0 & 1\end{bmatrix}} & (4)\end{matrix}$

-   -   A value of 1 in the BM matrix indicates a bus associated with        the line. The above mentioned system has two lines (Rows 1        and 2) and five buses (Columns 1-5). Line 1 (Row 1) has four        buses (Buses 1-4) represented by a value of one at Rows1-4. Line        2 (Row 2) has two buses (Bus 3 and 5) represented by a value of        one at Rows 3 and 5. A value of 0 indicates no bus at the        specific line (row).

-   2) S707, developing a new matrix called initial current matrix    (CM′).

$\begin{matrix}{{CM}^{\prime} = \begin{bmatrix}I_{11}^{\prime} & \ldots & I_{1N}^{\prime} \\\vdots & \ddots & \vdots \\I_{M1}^{\prime} & \ldots & I_{MN}^{\prime}\end{bmatrix}} & (4)\end{matrix}$

-   -   Where I′_(ij) represent the load current of Bus i given by this        formula:

$\begin{matrix}{I_{i\; j}^{\prime} = ( \frac{S_{i}}{V_{i}} )^{*}} & (5)\end{matrix}$

-   -   Where S_(i) represent complex power at Bus I, V_(i) represent        bus voltage. For the 5-bus test circuit displayed in FIG. 6, the        CM matrix is given as follows:

$\begin{matrix}{{C\; M^{\prime}} = \begin{bmatrix}I_{1}^{\prime} & I_{2}^{\prime} & I_{3}^{\prime} & I_{4}^{\prime} & 0 \\0 & 0 & I_{3}^{\prime} & 0 & I_{5}^{\prime}\end{bmatrix}} & (6)\end{matrix}$

-   3) In S709, the method determines the currents for the system    branches in a Zig Zag manner over the CM matrix and records them in    the final matrix denoted by CM matrix. Each bus will have its own    current tagged with the branch.

$\begin{matrix}{{CM} = \begin{bmatrix}I_{11} & \ldots & I_{1N} \\\vdots & \ddots & \vdots \\I_{M1} & \ldots & I_{MN}\end{bmatrix}} & (7) \\{I_{ij} = {\sum\limits_{n = 1}^{M}{\lbrack {\frac{I_{ij}^{\prime}}{K_{i}} +_{k}{\max\limits_{{k = 1},2,\ldots,N}I_{ik}}} \rbrack \times B_{ij}}}} & (8)\end{matrix}$

I_(ij) is called the Zig Zag formula which calculates the elements ofthe CM matrix in a Zig Zag manner. The symbol K_(i) reflects the numberof links connected to a specific Bus i (i.e., K is the number ofbranches from a source side in the case of multi-terminal lines).

-   -   The CM matrix for the 5-bus test circuit presented in FIG. 6 is        formulated as follows:

$\begin{matrix}{{C\; M} = \begin{bmatrix}I_{1} & I_{2} & I_{3} & I_{4} & 0 \\0 & 0 & I_{3} & 0 & I_{5}\end{bmatrix}} & (9)\end{matrix}$

-   4) In S711, the voltage will be identified using the CM matrix and    is calculated using the voltage drop formula, starting from upstream    through downstream.

The above steps are repeated iteratively until, in S713, the deviationbetween the load voltage and the previous voltage is below a specifiedtolerance for the process to converge. The tolerance may be apredetermined value that is within the precision of the computer systemused and the likely accuracy achievable by the load flow analysis.

When the Zig Zag-based method meets the convergence criteria (YES inS713), in S715 the computer system outputs final bus voltages V_(i). Insome embodiments, the final bus voltages may be displayed as a graph foreach bus, or a selected range of busses. From the bus voltages (angleand magnitude) all other power circuit parameters can be derived usingpower flow formulas. This includes active and reactive load flow,voltage drop, losses, etc.

FIG. 8 is a flowchart of a method for finding the currents for thebranches in a Zig Zag manner, in accordance with exemplary aspects ofthe disclosure.

The zig zag method is performed as follows:

-   -   1) In S801, initialize the current matrix (CM) by setting all        the elements to 0.    -   2) In S803, apply the Zig Zag formula developed above to        calculate all CM elements    -   3) In S805, in the current matrix (CM), perform vertical        movement: starting from 1×N (First Row×Last Column) moving to        Element M×N (Last Row×Last Column), where ‘x’ represents        position, adding the matrix element in the first row, Nth        column, to the next row in the Nth column, and continuing until        the last element in the last column is reached.

In S807, perform horizontal movement: move to the column before (N−1).

Repeating S803 to S807, working backwards through other columns, till,in S809, it completes the analysis for the whole matrix. In S811, the CMmatrix is saved and used in determining the voltages.

This Zig Zag methodology is applicable to extended radial withmulti-terminal lines distribution system. The matrices approach willcater for any extension in the system by expanding the CM dimension andreflecting the number of terminals in the K factor.

Results

Implementations of the BFS and Zig Zag methods have been developed andtested using 5, 7, 11, 25, 28, 30, 33, 34, and 69 - bus distributionsystems. The simulation results using the implementations are tabulatedin Table , Table 2 and Table 3.

TABLE 2 DEVIATION OF THE LAST TWO ITERATIONS (10⁻⁶) System BFS Zig Zag 5 5.61 2.85  7 2.69 2.69 11 2.95 2.95 25 7.75 4.10 28 9.53 6.45 30 9.352.13 33 7.93 0.22 34 9.23 3.63 69 7.87 1.44 Average 6.99 2.94

Accuracy of each of the methods is identified based on the differencesbetween last two iterations. Table 2 shows that the Zig Zag method issuperior in accuracy by 58% compared to the BFS method.

TABLE 1 Required Iterations to Convergence System BFS Zig Zag 5 9 7 7 88 11 9 9 25 42 7 28 141 23 30 44 7 33 25 4 34 46 7 69 50 7 Average 42 9

The average iterations required by the BFS method to converge isapproximately five times the iterations required by the Zig Zag methodas demonstrated in Table 1 (42 iterations for BFS compared to 9 for ZigZag). The number of iterations associated with each case study for thetwo methods are presented in Table 1. For example, the results for the5-bus test circuit converged after seven iterations for the Zig Zagmethod and resulted in a difference between the last iteration and theone before of 2.85E-06. This shows the robustness and the effectivenessof the Zig Zag method.

TABLE 3 Elapsed Time to Converge System BFS Zig Zag  5 0.024248 0.024681 7 0.020715 0.031041 11 0.021658 0.035335 25 0.027870 0.034460 280.039134 0.059562 30 0.029935 0.034840 33 0.028292 0.027398 34 0.0387840.032420 69 0.041999 0.049108 Average 0.030293 0.036538

In terms of elapsed time, though the Zig Zag method can providecomparatively outstanding results when compared to the BFS method, thetime taken by the Zig Zag method is pretty much the same as the BFSmethod as observed in Table 3, and also the quality of the solution isbetter. As a result, a conclusion can be made that the Zig Zag method isthe best technique for the studied systems and results in considerablybetter solutions. Also, better convergence time may be achieved from theZig Zag method by further optimization of the algorithm.

The voltage profiles of the simulated systems for 5, 7, 11, 25, 28, 30,33, 34, and 69 -bus distribution systems are displayed in FIG. 9 to FIG.17. The two methods are shown in each figure to compare the load flowresults. The figures prove the effectiveness of the Zig Zag method as ithas provided exactly the same voltage profiles as obtained by the BFSmethod.

Numerous modifications and variations of the present invention arepossible in light of the above teachings. It is therefore to beunderstood that within the scope of the appended claims, the inventionmay be practiced otherwise than as specifically described herein.

1-14. (canceled)
 15. A power distribution system including, inelectrical connection, an electricity generation plant, an electricitytransmission grid, and an electricity distribution grid having aplurality of power distribution lines, wherein the power distributionsystem is configured as a radial network comprising: a plurality ofbuses including a root bus having respective voltages; the plurality ofdistribution lines having respective currents and impedances, whereinthe radial network interconnects the plurality of buses with theplurality of distribution lines beginning with the root bus in atree-like manner, where the number of buses between a bus and the rootbus defines a layer, wherein subsets of the plurality of distributionlines are of different lengths, wherein a voltage of a bus at one of thelayers is different from a voltage of a bus at a different one of thelayers, wherein at least one of the buses has a link to at least two ofthe distribution lines, wherein the bus voltages and distribution linecurrents are determined by a processing circuitry configured to: receivea Branch Matrix (BM) having a row for each distribution line and acolumn for each bus, wherein the BM contains values indicating buses andassociated distribution lines, iteratively determine currents for theplurality of distribution lines and voltages across each of the busesuntil a difference between a bus voltage and a previous iterated busvoltage at each bus is below a predetermined tolerance and output finalbus voltages and final distribution line currents, and assign the finalvoltages to each respective bus and the final currents to eachrespective distribution line, wherein the circuitry iterativelydetermines the currents by determining an initial current matrix (CM)using the BM by multiplying a node current with corresponding matrixelements in the BM, wherein the node current is a ratio of a complexpower at a bus and a voltage of the bus, and determining the currentsfor the plurality of distribution lines in a zig zag manner over thematrix elements in the CM′ and records the determined currents in afinal current matrix (CM), the circuitry iteratively determines thevoltages by determining the voltage across each bus based on a voltagedrop between buses, starting from buses upstream and moving to busesdownstream in a direction of voltage flow.
 16. The system of claim 15,wherein the circuitry determines the currents for the plurality ofdistribution lines in the Zig Zag manner by iteratively performing, overall elements of the initial current matrix CM′, applying a Zig Zagformula to calculate CM elements I_(ij) in accordance with$I_{ij} = {\sum\limits_{n = 1}^{M}{\lbrack {\frac{I_{ij}^{\prime}}{K_{i}} +_{k}{\max\limits_{{k = 1},2,\ldots,N}I_{ik}}} \rbrack \times B_{ij}}}$wherein K₁ is a number of branches from a source side, B_(ij) representsthe existence of a bus in a certain branch, in a vertical movement,adding the matrix element in the first row, Nth column, to the next rowin the Nth column, and continuing until the last element in the lastrow, last column is reached; in a horizontal movement, moving to columnbefore N−1; and working backwards through all other columns in thematrix CM.
 17. The system of claim 15, wherein the circuitry determinesthe voltage across each bus based on a voltage drop between buses inaccordance withV _(i) =V _(j) −I _(ij) Z _(ij) where i is a specific bus, the symbol jis a bus with a former index (j<i) that is directly connected to i, andI_(ij) and Z_(ij) are the current and the impedance, respectively, ofthe distribution line between buses i and j.
 18. The system of claim 15,wherein the radial network is an extended radial network havingmulti-terminal distribution lines, wherein the number of links connectedto the bus is the number of multi-terminal distribution lines connectedto the bus.
 19. The system of claim 15, wherein the radial network hasan X/R ratio less than 10, wherein the X/R ratio is the amount ofreactance X divided by the amount of resistance R.
 20. The system ofclaim 15, wherein each bus is a bus bar, wherein the circuitryiteratively determines voltages across each bus bar.